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37\title{On the correctness of an assembler for the Intel MCS-51 microprocessor}
38\author{Dominic P. Mulligan \and Claudio Sacerdoti Coen\thanks{The project CerCo acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 243881}}
39\institute{Dipartimento di Scienze dell'Informazione, Universit\'a di Bologna}
40
41\bibliographystyle{splncs03}
42
43\begin{document}
44
45\maketitle
46
47\begin{abstract}
48We consider the formalisation of an assembler for Intel MCS-51 assembly language in the Matita proof assistant.
49This formalisation forms a major component of the EU-funded CerCo project, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
50
51The efficient expansion of pseudoinstructions---particularly jumps---into MCS-51 machine instructions is complex.
52We employ a strategy, involving the use of `policies', that separates the decision making over how jumps should be expanded from the expansion process itself.
53This makes the proof of correctness for the assembler significantly more straightforward.
54
55We prove, under the assumption of the existence of a correct policy, that the assembly process never fails and preserves the semantics of a subset of assembly programs.
56Correct policies fail to exist only in a limited number of pathological circumstances.
57Quite surprisingly, it is impossible for an optimising assembler to preserve the semantics of every assembly program.
58\end{abstract}
59
60% ---------------------------------------------------------------------------- %
61% SECTION                                                                      %
62% ---------------------------------------------------------------------------- %
63\section{Introduction}
64\label{sect.introduction}
65
66We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant~\cite{asperti:user:2007}.
67This formalisation forms a major component of the EU-funded CerCo project~\cite{cerco:2011}, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
68
69The MCS-51 dates from the early 1980s and is commonly called the 8051/8052.
70Despite the microprocessor's age, derivatives are still widely manufactured by a number of semiconductor foundries.
71As a result the processor is widely used, especially in embedded systems development, where well-tested, cheap, predictable microprocessors find their niche.
72
73The MCS-51 has a relative paucity of features compared to its more modern brethren.
74In particular, the MCS-51 does not possess a cache or any instruction pipelining that would make predicting the concrete cost of executing a single instruction an involved process.
75Instead, each semiconductor foundry that produces an MCS-51 derivative is able to provide accurate timing information in clock cycles for each instruction in their derivative's instruction set.
76It is important to stress that this timing information, unlike in more sophisticated processors, is not an estimate, it is a definition.
77For the MCS-51, if a manufacturer states that a particular opcode takes three clock cycles to execute, then that opcode \emph{always} takes three clock cycles to execute.
78
79This predicability of timing information is especially attractive to the CerCo consortium.
80We are in the process of constructing a certified, concrete complexity compiler for a realistic processor, and not for building and formalising the worst case execution time (WCET) tools that would be necessary to achieve the same result with, for example, a modern ARM or PowerPC microprocessor.
81
82However, the MCS-51's paucity of features is a double edged sword.
83In particular, the MCS-51 features relatively miniscule memory spaces (including read-only code memory, stack and internal/external random access memory) by modern standards.
84As a result our compiler, to have any sort of hope of successfully compiling realistic C programs, ought to produce `tight' machine code.
85This is not simple and requires the use of optimisations.
86
87For example, the MCS-51 features three unconditional jump instructions: \texttt{LJMP} and \texttt{SJMP}---`long jump' and `short jump' respectively---and an 11-bit oddity of the MCS-51, \texttt{AJMP}.
88Each of these three instructions expects arguments in different sizes and behaves in different ways: \texttt{SJMP} may only perform a `local jump'; \texttt{LJMP} may jump to any address in the MCS-51's memory space and \texttt{AJMP} may jump to any address in the current memory page.
89Consequently, the size of each opcode is different, and to squeeze as much code as possible into the MCS-51's limited code memory, the smallest possible opcode should be selected.
90
91The prototype CerCo C compiler does not attempt to select the smallest jump opcode in this manner, as this was thought to unneccessarily complicate the compilation chain.
92Instead, the compiler targets an assembly language, complete with pseudoinstructions including bespoke \texttt{Jmp} and \texttt{Call} instructions.
93Labels, conditional jumps to labels, a program preamble containing global data and a \texttt{MOV} instruction for moving this global data into the MCS-51's one 16-bit register also feature.
94This latter feature will ease any later consideration of separate compilation in the CerCo compiler.
95An assembler is used to expand pseudoinstructions into MCS-51 machine code.
96
97However, this assembly process is not trivial, for numerous reasons.
98For example, our conditional jumps to labels behave differently from their machine code counterparts.
99At the machine code level, all conditional jumps are `short', limiting their range.
100However, at the assembly level, conditional jumps may jump to a label that appears anywhere in the program, significantly liberalising the use of conditional jumps and further simplifying the design of the CerCo compiler.
101
102Further, trying to na\"ively relate assembly programs with their machine code counterparts simply does not work.
103Machine code programs that fetch from constant addresses in code memory or programs that combine the program counter with constant shifts do not make sense at the assembly level, since the position of instructions in code memory will be known only after assembly and optimisation.
104More generally, memory addresses can only be compared with other memory addresses.
105However, checking that memory addresses are only compared against each other at the assembly level is in fact undecidable.
106In short, we come to the shocking realisation that, with optimisations, the full preservation of the semantics of the two languages is impossible.
107We believe that this revelation is significant for large formalisation projects that assume the existence of a correct assembler.
108Projects in this class include both the recent CompCert~\cite{compcert:2011,leroy:formal:2009} and seL4 formalisations~\cite{klein:sel4:2009}.
109
110Yet, the situation is even more complex than having to expand pseudoinstructions correctly.
111In particular, when formalising the assembler, we must make sure that the assembly process does not change the timing characteristics of an assembly program for two reasons.
112First, the semantics of some functions of the MCS-51, notably I/O, depend on the microprocessor's clock.
113Changing how long a particular program takes to execute can affect the semantics of a program.
114This is undesirable.
115
116Second, as mentioned, the CerCo consortium is in the business of constructing a verified compiler for the C programming language.
117However, unlike CompCert~\cite{compcert:2011,leroy:formal:2009,leroy:formally:2009}---which currently represents the state of the art for `industrial grade' verified compilers---CerCo considers not just the \emph{extensional correctness} of the compiler, but also its \emph{intensional correctness}.
118That is, CompCert focusses solely on the preservation of the \emph{meaning} of a program during the compilation process, guaranteeing that the program's meaning does not change as it is gradually transformed into assembly code.
119However in any realistic compiler (even the CompCert compiler!) there is no guarantee that the program's time properties are preserved during the compilation process; a compiler's `optimisations' could, in theory, even conspire to degrade the concrete complexity of certain classes of programs.
120CerCo aims to expand the current state of the art by producing a compiler where this temporal degradation is guaranteed not to happen.
121Moreover, CerCo's approach lifts a program's timing information to the source (C language) level, wherein the programmer can reason about a program's intensional properties by directly examining the source code that they write.
122
123In order to achieve this CerCo imposes a cost model on programs, or more specifically, on simple blocks of instructions.
124This cost model is induced by the compilation process itself, and its non-compositional nature allows us to assign different costs to identical blocks of instructions depending on how they are compiled.
125In short, we aim to obtain a very precise costing for a program by embracing the compilation process, not ignoring it.
126This, however, complicates the proof of correctness for the compiler proper: for every translation pass from intermediate language to intermediate language, we must prove that not only has the meaning of a program been preserved, but also its complexity characteristics.
127This also applies for the translation from assembly language to machine code.
128
129How do we assign a cost to a pseudoinstruction?
130As mentioned, conditional jumps at the assembly level can jump to a label appearing anywhere in the program.
131However, at the machine code level, conditional jumps are limited to jumping `locally', using a measly byte offset.
132To translate a jump to a label, a single conditional jump pseudoinstruction may be translated into a block of three real instructions, as follows (here, \texttt{JZ} is `jump if accumulator is zero'):
133\begin{displaymath}
134\begin{array}{r@{\quad}l@{\;\;}l@{\qquad}c@{\qquad}l@{\;\;}l}
135       & \mathtt{JZ}  & label                      &                 & \mathtt{JZ}   & \text{size of \texttt{SJMP} instruction} \\
136       & \ldots       &                            & \text{translates to}   & \mathtt{SJMP} & \text{size of \texttt{LJMP} instruction} \\
137label: & \mathtt{MOV} & \mathtt{A}\;\;\mathtt{B}   & \Longrightarrow & \mathtt{LJMP} & \text{address of \textit{label}} \\
138       &              &                            &                 & \ldots        & \\
139       &              &                            &                 & \mathtt{MOV}  & \mathtt{A}\;\;\mathtt{B}
140\end{array}
141\end{displaymath}
142In the translation, if \texttt{JZ} fails, we fall through to the \texttt{SJMP} which jumps over the \texttt{LJMP}.
143Naturally, if \textit{label} is close enough, a conditional jump pseudoinstruction is mapped directly to a conditional jump machine instruction; the above translation only applies if \textit{label} is not sufficiently local.
144This leaves the problem, addressed below, of calculating whether a label is indeed `close enough' for the simpler translation to be used.
145
146Crucially, the above translation demonstrates the difficulty in predicting how many clock cycles a pseudoinstruction will take to execute.
147A conditional jump may be mapped to a single machine instruction or a block of three.
148Perhaps more insidious, the number of cycles needed to execute the instructions in the two branches of a translated conditional jump may be different.
149Depending on the particular MCS-51 derivative at hand, an \texttt{SJMP} could in theory take a different number of clock cycles to execute than an \texttt{LJMP}.
150These issues must also be dealt with in order to prove that the translation pass preserves the concrete complexity of the code, and that the semantics of a program using the MCS-51's I/O facilities does not change. We will address this
151problem parameterizing the semantics over a cost model. We will prove preservation of concrete complexity only for the program-dependent cost model induced by the optimization.
152
153The question remains: how do we decide whether to expand a jump into an \texttt{SJMP} or an \texttt{LJMP}?
154To understand why this problem is not trivial, consider the following snippet of assembly code:
155\begin{displaymath}
156\text{dpm: finish me}
157\end{displaymath}
158
159As our example shows, given an occurence $l$ of an \texttt{LJMP} instruction, it may be possible to shrink $l$ to an occurence of an \texttt{SJMP} providing we can shrink any \texttt{LJMP}s that exist between $l$ and its target location.
160However, shrinking these \texttt{LJMP}s may in turn depend on shrinking $l$ to an \texttt{SJMP}, as it is perfectly possible to jump backwards.
161In short, unless we can somehow break this loop of circularity, and similar knotty configurations of jumps, we are stuck with a suboptimal solution to the expanding jumps problem.
162
163How we went about resolving this problem affected the shape of our proof of correctness for the whole assembler in a rather profound way.
164We first attempted to synthesize a solution bottom up: starting with no solution, we gradually refine a solution using the same functions that implement the jump expansion.
165Using this technique, solutions can fail to exist, and the proof quickly descends into a diabolical quagmire.
166
167Abandoning this attempt, we instead split the `policy'---the decision over how any particular jump should be expanded---from the implementation that actually expands assembly programs into machine code.
168Assuming the existence of a correct policy, we proved the implementation of the assembler correct.
169Further, we proved that the assembler fails to assemble an assembly program if and only if a correct policy does not exist.
170Policies do not exist in only a limited number of circumstances: namely, if a pseudoinstruction attempts to jump to a label that does not exist, or the program is too large to fit in code memory.
171The first case would constitute a serious compiler error, and hopefully certifying the rest of the compiler would rule this possibility out.
172The second case is unavoidable---certified compiler or not, trying to load a huge program into a small code memory will break \emph{something}.
173
174The rest of this paper is a detailed description of this proof.
175
176% ---------------------------------------------------------------------------- %
177% SECTION                                                                      %
178% ---------------------------------------------------------------------------- %
179\subsection{Overview of the paper}
180\label{subsect.overview.of.the.paper}
181In Section~\ref{sect.matita} we provide a brief overview of the Matita proof assistant for the unfamiliar reader.
182In Section~\ref{sect.the.proof} we discuss the design and implementation of the proof proper.
183In Section~\ref{sect.conclusions} we conclude.
184
185% ---------------------------------------------------------------------------- %
186% SECTION                                                                      %
187% ---------------------------------------------------------------------------- %
188\section{Matita}
189\label{sect.matita}
190
191Matita is a proof assistant based on the Calculus of (Co)inductive Constructions~\cite{asperti:user:2007}.
192For those familiar with Coq, Matita's syntax and mode of operation should be entirely familiar.
193However, we take time here to explain one of Matita's syntactic idiosyncrasies.
194The use of `$\mathtt{?}$' or `$\mathtt{\ldots}$' in an argument position denotes a term or terms to be inferred automatically by unification, respectively.
195The use of `$\mathtt{?}$' in the body of a definition, lemma or axiom denotes an incomplete term that is to be closed, by hand, using tactics.
196
197% ---------------------------------------------------------------------------- %
198% SECTION                                                                      %
199% ---------------------------------------------------------------------------- %
200\section{The proof}
201\label{sect.the.proof}
202
203\subsection{The assembler and semantics of machine code}
204\label{subsect.the.assembler.and.semantics.of.machine.code}
205
206The formalisation in Matita of the semantics of MCS-51 machine code is described in~\cite{mulligan:executable:2011}.
207We merely describe enough here to understand the rest of the proof.
208
209At heart, the MCS-51 emulator centres around a \texttt{Status} record, describing the current state of the microprocessor.
210This record contains fields corresponding to the microprocessor's program counter, special function registers, and so on.
211At the machine code level, code memory is implemented as a trie of bytes, addressed by the program counter.
212Machine code programs are loaded into \texttt{Status} using the \texttt{load\_code\_memory} function.
213
214We may execut a single step of a machine code program using the \texttt{execute\_1} function, which returns an updated \texttt{Status}:
215\begin{lstlisting}
216definition execute_1: Status $\rightarrow$ Status := $\ldots$
217\end{lstlisting}
218The function \texttt{execute} allows one to execute an arbitrary, but fixed (due to Matita's normalisation requirement!) number of steps of a program.
219
220Naturally, assembly programs have analogues.
221The counterpart of the \texttt{Status} record is \texttt{PseudoStatus}.
222Instead of code memory being implemented as tries of bytes, code memory is here implemented as lists of pseudoinstructions, and program counters are merely indices into this list.
223Our analogue of \texttt{execute\_1} is \texttt{execute\_1\_pseudo\_instruction}:
224\begin{lstlisting}
225definition execute_1_pseudo_instruction: (Word $\rightarrow$ nat $\times$ nat) $\rightarrow$
226                                         PseudoStatus $\rightarrow$ PseudoStatus := $\ldots$
227\end{lstlisting}
228Notice, here, that the emulation function for assembly programs takes an additional argument.
229This is a function that maps program counters (at the assembly level) to pairs of natural numbers representing the number of clock ticks that the pseudoinstruction needs to execute, post expansion.
230We call this function a \emph{costing}, and note that the costing is induced by the particular strategy we use to expand pseudoinstructions.
231If we change how we expand conditional jumps to labels, for instance, then the costing needs to change, hence \texttt{execute\_1\_pseudo\_instruction}'s parametricity in the costing.
232
233The costing returns \emph{pairs} of natural numbers because, in the case of expanding conditional jumps to labels, the expansion of the `true branch' and `false branch' may differ in the number of clock ticks needed for execution.
234This timing information is used inside \texttt{execute\_1\_pseudo\_instruction} to update the clock of the \texttt{PseudoStatus}.
235During the proof of correctness of the assembler we relate the clocks of \texttt{Status}es and \texttt{PseudoStatus}es.
236
237The assembler, mapping programs consisting of lists of pseudoinstructions to lists of bytes, operates in a mostly straightforward manner.
238To a degree of approximation, the assembler on an assembly program, consisting of $n$ pseudoinstructions $\mathtt{P_i}$ for $1 \leq i \leq n$, works as in the following diagram:
239\begin{displaymath}
240[\mathtt{P_1}, \ldots \mathtt{P_n}] \xrightarrow{\mathtt{flatten}\left(\mathtt{P_i} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{expand}$}} \mathtt{[I_1^i, \ldots I^q_i]} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{assembly1}^*$}} \mathtt{[0110]}\right)^{*}} \mathtt{[010101]}
241\end{displaymath}
242Here $\mathtt{I^i_j}$ for $1 \leq j \leq q$ are the $q$ machine code instructions obtained by expanding, with \texttt{expand\_pseudo\_instruction}, a single pseudoinstruction.
243Each machine code instruction $\mathtt{I^i_j}$ is then assembled, using the \texttt{assembly1} function, into a list of bytes.
244This process is iterated for each pseudoinstruction, before the lists are flattened into a single bit list representation of the original assembly program.
245
246By inspecting the above diagram, it would appear that the best way to proceed with a proof that the assembly process does not change the semantics of an assembly program is via a decomposition of the problem into two subproblems.
247Namely, we first expand any and all pseudoinstructions into lists of machine instructions, and provide a proof that this process does not change our program's semantics.
248Finally, we assemble all machine code instructions into machine code---lists of bytes---and prove once more that this process does not have an adverse effect on a program's semantics.
249By composition, we then have that the whole assembly process is semantics preserving.
250
251This is a tempting approach to the proof, but ultimately the wrong approach.
252In particular, it is important that we track how the program counter indexing into the assembly program, and the machine's program counter evolve, so that we can relate them.
253Expanding pseudoinstructions requires that the machine's program counter be incremented by $n$ steps, for $1 \leq n$, for every increment of the assembly program's program counter.
254Keeping track of the correspondence between the two program counters quickly becomes unfeasible using a compositional approach, and hence the proof must be monolithic.
255
256% ---------------------------------------------------------------------------- %
257% SECTION                                                                      %
258% ---------------------------------------------------------------------------- %
259\subsection{Policies}
260\label{subsect.policies}
261
262Policies exist to dictate how conditional and unconditional jumps at the assembly level should be expanded into machine code instructions.
263Using policies, we are able to completely decouple the decision over how jumps are expanded with the act of expansion, simplifying our proofs.
264As mentioned, the MCS-51 instruction set includes three different jump instructions: \texttt{SJMP}, \texttt{AJMP} and \texttt{LJMP}; call these `short', `medium' and `long' jumps, respectively:
265\begin{lstlisting}
266inductive jump_length: Type[0] :=
267  | short_jump: jump_length
268  | medium_jump: jump_length
269  | long_jump: jump_length.
270\end{lstlisting}
271A \texttt{jump\_expansion\_policy} is a map from \texttt{Word}s to \texttt{jump\_length}s, implemented as a trie.
272Intuitively, a policy maps positions in a program (indexed using program counters implemented as \texttt{Word}s) to a particular variety of jump.
273\begin{lstlisting}
274definition jump_expansion_policy := BitVectorTrie jump_length 16.
275\end{lstlisting}
276Next, we require a series of `sigma' functions.
277These functions map assembly program counters to their machine code counterparts, establishing the correspondence between `positions' in an assembly program and `positions' in a machine code program.
278At the heart of this process is \texttt{sigma0} which traverses an assembly program building maps from program counter to program counter.
279This function fails if and only if an internal call to \texttt{assembly\_1\_pseudoinstruction} fails:
280\begin{lstlisting}
281definition sigma0: pseudo_assembly_program
282  $\rightarrow$ option (nat $\times$ (nat $\times$ (BitVectorTrie Word 16))) := $\ldots$
283\end{lstlisting}
284We eventually lift this to functions from program counters to program counters:
285\begin{lstlisting}
286definition sigma_safe:
287  pseudo_assembly_program $\rightarrow$ option (Word $\rightarrow$ Word) := $\ldots$
288\end{lstlisting}
289Now, it's possible to define what a `good policy' is i.e. one that does not cause \texttt{sigma\_safe} to fail.
290As mentioned, \texttt{sigma\_safe} can only fail if an assembly program fails to be assembled:
291\begin{lstlisting}
292definition policy_ok := $\lambda$p. sigma_safe p $\neq$ None $\ldots$.
293\end{lstlisting}
294Finally, we obtain \texttt{sigma}, a map from program counters to program counters, which is guranteed not to fail as we internally provide a that
295\begin{lstlisting}
296definition sigma: pseudo_assembly_program $\rightarrow$ Word $\rightarrow$ Word := $\ldots$
297\end{lstlisting}
298
299% ---------------------------------------------------------------------------- %
300% SECTION                                                                      %
301% ---------------------------------------------------------------------------- %
302\subsection{Total correctness of the assembler}
303\label{subsect.total.correctness.of.the.assembler}
304
305Using our policies, we now work toward proving the total correctness of the assembler.
306By total correctness, we mean that the assembly process does not change the semantics of an assembly program.
307Naturally, this necessitates keeping some sort of correspondence between program counters at the assembly level, and program counters at the machine code level.
308For this, we use the \texttt{sigma} machinery defined at the end of Subsection~\ref{subsect.policies}.
309
310We expand pseudoinstructions using the function \texttt{expand\_pseudo\_instruction}.
311This function accepts a `policy decision'---an element of type \texttt{jump\_length}---that is used when expanding a \texttt{Call}, \texttt{Jmp} or conditional jump to a label into the correct machine instruction.
312This \texttt{policy\_decision} is asssumed to originate from a policy as defined in Subsection~\ref{subsect.policies}.
313\begin{lstlisting}
314definition expand_pseudo_instruction:
315  ∀lookup_labels, lookup_datalabels, pc, policy_decision.
316    pseudo_instruction $\rightarrow$ option list instruction := $\ldots$
317\end{lstlisting}
318Under the assumption that a correct policy exists, \texttt{expand\_pseudo\_instruction} should never fail, and therefore the option type may be dispensed with.
319This is because the only failure conditions for \texttt{expand\_pseudo\_instruction} result from trying to expand a pseudoinstruction into an `impossible' combination of machine code instructions.
320For instance, if the policy decision dictates that we should expand a \texttt{Call} pseudoinstruction into a `short jump', then we fail, as the MCS-51's instruction set only features instructions \texttt{ACALL} and \texttt{LCALL}.
321
322% dpm todo
323\begin{lstlisting}
324axiom assembly_ok: ∀program,assembled,costs,labels.
325  Some $\ldots$ $\langle$labels, costs$\rangle$ = build_maps program $\rightarrow$
326  Some $\ldots$ $\langle$assembled, costs$\rangle$ = assembly program $\rightarrow$
327  let code_memory := load_code_memory assembled in
328  let preamble := $\pi_1$ program in
329  let datalabels := construct_datalabels preamble in
330  let lk_labels :=
331    $\lambda$x. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
332  let lk_dlabels := $\lambda$x. lookup ? ? x datalabels (zero ?) in
333   ∀ppc,len,assembledi.
334    let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
335    let assembly' := assembly_1_pseudoinstruction program ppc
336      (sigma program ppc) lk_labels lk_dlabels pi in
337    let newpc := (sigma program ppc) + len in
338    let echeck :=
339      encoding_check code_memory (sigma program ppc) slen assembledi in
340     Some $\ldots$ $\langle$len, assembledi$\rangle$ = assembly' $\rightarrow$
341      echeck $\wedge$ sigma program newppc = newpc.
342\end{lstlisting}
343
344% dpm todo
345\begin{lstlisting}
346theorem fetch_assembly: $\forall$pc, i, cmem, assembled.
347  assembled = assembly1 i $\rightarrow$
348  let len := length $\ldots$ assembled in
349    encoding_check cmem pc (pc + len) assembled $\rightarrow$
350    let fetched := fetch code_memory (bitvector_of_nat $\ldots$ pc) in
351    let $\langle$instr_pc, ticks$\rangle$ := fetched in
352    let $\langle$instr, pc'$\rangle$ := instr_pc in
353      (eq_instruction instr i $\wedge$
354       eqb ticks (ticks_of_instruction instr) $\wedge$
355       eq_bv $\ldots$ pc' (pc + len)) = true.
356\end{lstlisting}
357
358Lemma \texttt{fetch\_assembly\_pseudo} establishes a basic relationship between \texttt{expand\_pseudo\_instruction} and \texttt{assembly\_1\_pseudoinstruction}:
359\begin{lstlisting}
360lemma fetch_assembly_pseudo: $\forall$program, ppc, lk_labels, lk_dlabels.
361  $\forall$pi, code_memory, len, assembled, instructions, pc.
362  let jexp := jump_expansion ppc program in
363  let exp :=
364    expand_pseudo_instruction lk_labels lk_dlabels pc jexp pi
365  let ass :=
366    assembly_1_pseudoinstruction program ppc pc lk_labels lk_dlabels pi in
367  Some ? instructions = exp $\rightarrow$
368    Some $\ldots$ $\langle$len, assembled$\rangle$ = ass $\rightarrow$
369      encoding_check code_memory pc (pc + len) assembled $\rightarrow$
370        fetch_many code_memory (pc + len) pc instructions.
371\end{lstlisting}
372Here, \texttt{len} is the number of machine code instructions the pseudoinstruction at hand has been expanded into, \texttt{encoding\_check} is a recursive function that checks for any possible corruption of the code memory, resulting from expanding the pseudoinstruction.
373We assemble a single pseudoinstruction with \texttt{assembly\_1\_pseudoinstruction}, which internally calls \texttt{jump\_expansion} and \texttt{expand\_pseudo\_instruction}.
374The function \texttt{fetch\_many} fetches multiple machine code instructions from code memory and performs some routine checks.
375
376Intuitively, Lemma \texttt{fetch\_assembly\_pseudo} can be read as follows.
377Suppose our policy \texttt{jump\_expansion} dictates that the pseudoinstruction indexed by the pseudo program counter \texttt{ppc} in assembly program \texttt{program} gives us the policy decision \texttt{jexp}.
378Further, suppose we expand the pseudoinstruction at \texttt{ppc} with the policy decision \texttt{jexp}, obtaining an (optional) list of machine code instructions \texttt{exp}.
379Suppose we also assemble the pseudoinstruction at \texttt{ppc} to obtain \texttt{ass}, a list of bytes.
380Then, under the assumption that neither the expansion of the pseudoinstruction to obtain \texttt{exp}, nor the assembly of the pseudoinstruction to obtain \texttt{ass}, failed, we check with \texttt{fetch\_many} that the number of machine instructions that were fetched matches the number of instruction that \texttt{expand\_pseudo\_instruction} expanded.
381
382At first sight, Lemma \texttt{fetch\_assembly\_pseudo2} appears to nearly establish the correctness of the assembler:
383\begin{lstlisting}
384lemma fetch_assembly_pseudo2: $\forall$program, assembled, costs, labels.
385  Some $\ldots$ $\langle$labels, costs$\rangle$ = build_maps program $\rightarrow$
386  Some $\ldots$ $\langle$assembled, costs$\rangle$ = assembly program $\rightarrow$ $\forall$ppc.
387  let code_memory := load_code_memory assembled in
388  let preamble := $\pi_1$ program in
389  let data_labels := construct_datalabels preamble in
390  let lk_labels :=
391    λx. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
392  let lk_dlabels := λx. lookup ? ? x data_labels (zero ?) in
393  let expansion := jump_expansion ppc program in
394  let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
395  let ppc' := sigma program ppc in
396  let newppc' := sigma program newppc in
397  let instructions' :=
398    expand_pseudo_instruction lk_labels lk_dlabels ppc' expansion pi in
399  let fetched := $\lambda$instr. fetch_many code_memory newppc' ppc' instr in
400    $\exists$instrs. Some ? instrs = instructions' $\wedge$ fetched instrs.
401\end{lstlisting}
402Intuitively, we may read \texttt{fetch\_assembly\_pseudo2} as follows.
403Suppose we are able to successfully assemble an assembly program using \texttt{assembly} and produce a code memory, \texttt{code\_memory}.
404Then there exists some list of machine instructions equal to the expansion of a pseudoinstruction and the number of machine instructions that need to be fetched is equal to the number of machine instructions that the pseudoinstruction was expanded into.
405
406However, this property is \emph{not} strong enough to establish that the semantics of an assembly program has been preserved by the assembly process.
407In particular, \texttt{fetch\_assembly\_pseudo2} says nothing about how memory addresses evolve during assembly.
408Memory addresses in one memory space may be mapped to memory addresses in a completely different memory space during assembly.
409To handle this problem, we need some more machinery.
410
411We use an \texttt{internal\_pseudo\_address\_map} for this purpose.
412An \texttt{internal\_pseudo\_address\_map} associates positions in the memory of a \texttt{PseudoStatus} with a physical memory address:
413\begin{lstlisting}
414definition internal_pseudo_address_map := list (BitVector 8).
415\end{lstlisting}
416We use \texttt{internal\_pseudo\_address\_map}s to convert the lower internal RAM of a \texttt{PseudoStatus} into the lower internal RAM of a \texttt{Status}.
417The actual conversion process is performed by \texttt{low\_internal\_ram\_of\_pseudo\_low\_internal\_ram}:\footnote{An associated set of axioms describe how \texttt{low\_internal\_ram\_of\_pseudo\_low\_internal\_ram} behaves.  This is a form of parametricity.  We don't care about the particulars of the conversion functions, as long as they behave in accordance with our axioms.}
418
419\begin{lstlisting}
420axiom low_internal_ram_of_pseudo_low_internal_ram:
421  internal_pseudo_address_map $\rightarrow$ BitVectorTrie Byte 7 $\rightarrow$ BitVectorTrie Byte 7.
422\end{lstlisting}
423A similar axiom exists for high internal RAM.
424
425Notice, the MCS-51's internal RAM is addressed with a 7-bit `byte'.
426% dpm: ugly English, fix
427The whole of the internal RAM space is addressed with bytes: the first bit is used to distinguish between the programmer addressing low and high internal memory.
428
429Next, we are able to translate \texttt{PseudoStatus} records into \texttt{Status} records using \texttt{status\_of\_pseudo\_status}.
430Translating a \texttt{PseudoStatus}'s code memory requires we expand pseudoinstructions and then assemble to obtain a trie of bytes.
431This can fail, as mentioned, in a limited number of situations, related to improper use of labels in an assembly program.
432However, it is possible to `tighten' the type of \texttt{status\_of\_pseudo\_status}, removing the option type, by using the fact that if any `good policy' exists, assembly will never fail.
433\begin{lstlisting}
434definition status_of_pseudo_status:
435 internal_pseudo_address_map → PseudoStatus → option Status
436\end{lstlisting}
437After fetching an assembly instruction we must update any \texttt{internal\_pseudo\hyp{}\_address\_map}s that may be laying around.
438This is done with the following function:
439\begin{lstlisting}
440definition next_internal_pseudo_address_map: internal_pseudo_address_map
441  $\rightarrow$ PseudoStatus $\rightarrow$ option internal_pseudo_address_map
442\end{lstlisting}
443Finally, we are able to state and prove our main theorem.
444This relates the execution of a single assembly instruction and the execution of (possibly) many machine code instructions.
445That is, the assembly process preserves the semantics of an assembly program, as it is translated into machine code:
446\begin{lstlisting}
447theorem main_thm:
448  ∀M,M',ps,s,s''.
449    next_internal_pseudo_address_map M ps = Some $\ldots$ M' $\rightarrow$
450      status_of_pseudo_status M ps = Some $\ldots$ s $\rightarrow$
451        status_of_pseudo_status M'
452          (execute_1_pseudo_instruction
453            (ticks_of (code_memory $\ldots$ ps)) ps) = Some $\ldots$ s'' $\rightarrow$
454              $\exists$n. execute n s = s''.
455\end{lstlisting}
456The statement can be given an intuitive reading as follows.
457Suppose our \texttt{PseudoStatus}, \texttt{ps}, can be successfully converted into a \texttt{Status}, \texttt{s}.
458Suppose further that, after executing a single assembly instruction and converting the resulting \texttt{PseudoStatus} into a \texttt{Status}, we obtain \texttt{s''}, being careful to track the number of ticks executed.
459Then, there exists some number \texttt{n}, so that executing \texttt{n} machine code instructions in \texttt{Status} \texttt{s} gives us \texttt{Status} \texttt{s''}.
460Theorem \texttt{main\_thm} establishes the correctness of the assembly process.
461
462% ---------------------------------------------------------------------------- %
463% SECTION                                                                      %
464% ---------------------------------------------------------------------------- %
465\section{Conclusions}
466\label{sect.conclusions}
467
468We have proved the total correctness of an assembler for MCS-51 assembly language.
469In particular, our assembly language featured labels, arbitrary conditional and unconditional jumps to labels, global data and instructions for moving this data into the MCS-51's single 16-bit register.
470Expanding these pseudoinstructions into machine code instructions is not trivial, and the proof that the assembly process is `correct', in that the semantics of an assembly program are not changed is complex.
471
472The formalisation is a key component of the CerCo project, which aims to produce a verified concrete complexity preserving compiler for a large subset of the C programming language.
473The verified assembler, complete with the underlying formalisation of the semantics of MCS-51 machine code (described fully in~\cite{mulligan:executable:2011}), will form the bedrock layer upon which the rest of the CerCo project will build its verified compiler platform.
474
475Aside from their application in verified compiler projects such as CerCo, verified assemblers such as ours could also be applied to the verification of operating system kernels.
476Of particular note is the verified seL4 kernel~\cite{klein:sel4:2009,klein:sel4:2010}.
477This verification explicitly assumes the existence of, amongst other things, a trustworthy assembler and compiler.
478
479\paragraph{Use of dependent types and Russell}
480Our formalisation makes sparing use of dependent types.
481In certain datastructures, such as tries and vectors, they are used to guarantee invariants.
482However, we have currently shyed away from making extensive use of dependent types and inductive predicates in the proof of correctness for the assembler itself.
483This is because complex dependent types and inductive predicates tend not to co\"operate particularly well with tactics such as inversion.
484
485However, there are certain cases where the use of dependent types is unavoidable.
486For instance, when proving that the \texttt{build\_maps} function is correct, a function that collates the cost and data labels of an assembly program into map datastructures.
487In cases such as these we make use of Matita's implementation of Russell~\cite{sozeau:subset:2006}.
488In Matita, Russell may be implemented with two coercions and some notational sugaring.
489
490\subsection{Related work}
491\label{subsect.related.work}
492
493% piton
494We are not the first to consider the total correctness of an assembler for a non-trivial assembly language.
495Perhaps the most impressive piece of work in this domain is the Piton stack~\cite{moore:piton:1996,moore:grand:2005}.
496This was a stack of verified components, written and verified in ACL2, ranging from a proprietary FM9001 microprocessor verified at the gate level, to assemblers and compilers for two high-level languages---a dialect of Lisp and $\mu$Gypsy~\cite{moore:grand:2005}.
497%dpm more: weirich paper?
498
499% jinja
500Klein and Nipkow consider a Java-like programming language, Jinja~\cite{klein:machine:2006,klein:machine:2010}.
501They provide a compiler, virtual machine and operational semantics for the programming language and virtual machine, and prove that their compiler is semantics and type preserving.
502
503Finally, mention should be made of CompCert~\cite{compcert:2011,blazy:formal:2006,leroy:formal:2009,leroy:formally:2009}, another verified compiler project related to CerCo.
504As previously mentioned, CompCert considers only extensional correctness of the compiler, and not intensional correctness, which CerCo focusses on.
505However, CerCo also extends CompCert in other ways.
506Namely, the CompCert verified compilation chain terminates at the PowerPC or ARM assembly level, and takes for granted the existence of a trustworthy assembler.
507CerCo chooses to go further, by considering a verified compilation chain all the way down to the machine code level.
508In essence, the work presented in this publication is one part of CerCo's extension over CompCert.
509
510\bibliography{cpp-2011.bib}
511
512\end{document}\renewcommand{\verb}{\lstinline}
513\def\lstlanguagefiles{lst-grafite.tex}
514\lstset{language=Grafite}
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